Fixed Point Iteration Calculator
Work out fixed point iteration instantly with clear inputs, formula shown and shareable results.
About Fixed Point Iteration Calculator
Fixed-point iteration repeatedly applies x ← g(x) until the value settles. The Banach condition |g'(x)| < 1 guarantees convergence, and smaller values converge faster.
Formulas
Fixed point iteration
xₙ₊₁ = g(xₙ), converging when |g'(x)| < 1
Linear case
fixed point = b / (1 − a)
Worked example
With the default values below, Fixed Point Iteration Calculator returns Fixed point reached of 5.9999952316. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Coefficient a in g(x) = a·x + b
- 0.5
- Constant b
- 3
- Starting value
- 1
- Iterationsaccepts 1–500
- 20
Results
- Fixed point reached
- 5.9999952316
- Exact fixed point b/(1 − a)
- 6.0000000000
- Convergence condition
- Converges — |g'(x)| = 0.5 is below 1
- Iterations used
- 20
Step by step
- 1Values used
- 2Fixed point iteration
- 3Linear case
- 4Fixed point reached= 5.9999952316
- 5Exact fixed point b/(1 − a)= 6.0000000000
- 6Convergence condition= Converges — |g'(x)| = 0.5 is below 1
- 7Iterations used= 20
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
For g(x) = 0.5x + 3 from x = 1?
It converges to 3/(1 − 0.5) = 6.
Why does |g'| < 1 matter?
Each step multiplies the error by roughly g', so a value below 1 shrinks the error and above 1 amplifies it.